| L(s) = 1 | + 5.05·2-s + 17.5·4-s + 5·5-s + 19.3·7-s + 48.0·8-s + 25.2·10-s − 6.81·11-s + 36.9·13-s + 97.9·14-s + 102.·16-s + 71.5·17-s + 88.3·19-s + 87.5·20-s − 34.4·22-s − 185.·23-s + 25·25-s + 186.·26-s + 339.·28-s − 29·29-s − 120.·31-s + 133.·32-s + 361.·34-s + 96.9·35-s − 117.·37-s + 445.·38-s + 240.·40-s + 229.·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.18·4-s + 0.447·5-s + 1.04·7-s + 2.12·8-s + 0.798·10-s − 0.186·11-s + 0.788·13-s + 1.86·14-s + 1.60·16-s + 1.02·17-s + 1.06·19-s + 0.978·20-s − 0.333·22-s − 1.68·23-s + 0.200·25-s + 1.40·26-s + 2.29·28-s − 0.185·29-s − 0.697·31-s + 0.736·32-s + 1.82·34-s + 0.468·35-s − 0.521·37-s + 1.90·38-s + 0.948·40-s + 0.874·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.039505933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.039505933\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.05T + 8T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 11 | \( 1 + 6.81T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 66.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 9.42T + 1.48e5T^{2} \) |
| 59 | \( 1 - 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 953.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 429.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 554.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 965.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 625.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.387107932630598617799241810789, −8.097821472989554184082380410203, −7.51626309022680249450619557916, −6.41535057262501688224037579848, −5.60636683820418129333486749737, −5.19677427578860267863209889746, −4.12863180188214425155256905578, −3.39574750201322704650397877823, −2.23978371928515366652093229612, −1.34101284007026192823134828260,
1.34101284007026192823134828260, 2.23978371928515366652093229612, 3.39574750201322704650397877823, 4.12863180188214425155256905578, 5.19677427578860267863209889746, 5.60636683820418129333486749737, 6.41535057262501688224037579848, 7.51626309022680249450619557916, 8.097821472989554184082380410203, 9.387107932630598617799241810789
